Linear systems on tropical curves

Haase, Christian; Musiker, Gregg; Yu, Josephine

doi:10.1007/s00209-011-0844-4

Cited by 54 publications

(73 citation statements)

References 15 publications

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“…We define an abstract tropical curve in terms of star-shaped sets, as a generalization of a metric (or metrized) graph in which all leaves have infinite length. Our definition is based on papers of Zhang (1993), Baker and Rumely (2007), and Haase et al (2009). See also Mikhalkin and Zharkov (2008), Mikhalkin (2006), and Baker and Faber (2006) (Fig.…”

Section: Background On Abstract Tropical Curvesmentioning

confidence: 99%

“…Our definition is based on papers of Zhang (1993), Baker and Rumely (2007), and Haase et al (2009). See also Mikhalkin and Zharkov (2008), Baker and Faber (2006), and Richter-Gebert et al (2005).…”

Section: Introductionmentioning

confidence: 98%

“…See also Mikhalkin and Zharkov (2008), Baker and Faber (2006), and Richter-Gebert et al (2005). We define rational functions, divisors, and divisor classes in this setting, following the conventions of Mikhalkin and Zharkov (2008), Gathmann and Kerber (2008), and Haase et al (2009). We note that the automorphism group of an abstract tropical curve X is necessarily finite unless X is homeomorphic to a circle or a closed interval.…”

Section: Introductionmentioning

confidence: 99%

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Automorphism groups on tropical curves: some cohomology calculations

2011

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Let X be an abstract tropical curve and let G be a finite subgroup of the automorphism group of X . Let D be a divisor on X whose equivalence class is G-invariant. We address the following question: is there always a divisor D in the equivalence class of D which is G-invariant? Our main result is that the answer is "yes" for all abstract tropical curves. A key step in our proof is a tropical analogue of Hilbert's Theorem 90.

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Section: Background On Abstract Tropical Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Automorphism groups on tropical curves: some cohomology calculations

2011

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“…However, this approach also faces significant difficulties. See [HMY12] for a detailed discussion of the tropical semimodule structure on R(D).…”

Section: Jacobians Of Metric Graphsmentioning

confidence: 99%

Degeneration of Linear Series from the Tropical Point of View and Applications

Baker

Jensen

2016

Simons Symposia

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“…primary 05C30, secondary 05C25. For the combinatorial structure of |D| in the case of a metric graph (tropical curve), see [13]. 1…”

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confidence: 99%

Enumerating linear systems on graphs

2020

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The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D-the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. The final section generalizes our results to a model based on integral M -matrices.2010 Mathematics Subject Classification. primary 05C30, secondary 05C25. For the combinatorial structure of |D| in the case of a metric graph (tropical curve), see [13]. 1 Divisor theory preliminariesLet G = (V, E) be a connected, undirected multigraph with finite vertex set V and finite edge multiset E. Many of our constructions will depend on fixing a vertex q ∈ V , which we do now, once and for all. Loops are allowed but our results are not affected if they are removed. We let N := Z ≥0 denote the natural numbers.We recall some of the theory of divisors on graphs, referring readers unfamiliar with this theory to [3] or to the textbooks [8] and [15]. A divisor on G is an element of the free abelian group on the vertices of G,The degree of a divisor D is the sum of its coefficients: deg(D) := v∈V D(v). For instance, if we consider v ∈ V as a divisor, then deg(v) = 1. We use the notation deg G (v) to refer to the ordinary degree of a vertex-the number of edges incident on v. The set of divisors of degree k is denoted by Div k (G). The (discrete) Laplacian operator of G is the function L : Z V → Z V given by

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Linear systems on tropical curves

Cited by 54 publications

References 15 publications

Automorphism groups on tropical curves: some cohomology calculations

Automorphism groups on tropical curves: some cohomology calculations

Degeneration of Linear Series from the Tropical Point of View and Applications

Enumerating linear systems on graphs

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